Did You Find Out? Get the Answer to the Riddle of E-News Issue No. 5, September 2015

Does Mrs Müller have a son and a daughter?

Does Mrs Müller have a son and a daughter?

Mrs Müller has well-kept girls clothes that no longer fit her daughter. In the neighborhood, Mrs Schmidt lives with her two little children. Mrs Müller would like to give Mrs Schmidt the clothing if she could use it. Thus, Mrs Müller asks whether Mrs Schmidt has a daughter and gets the information that yes, she does. As Mrs Müller’s friends heard about her interest in donating clothing, they also provide their children‘s clothing which they no longer need. These are both girls and boys clothes. Mrs Müller starts to wonder whether Mrs Schmidt has two daughters or if she has a son and a daugher?

What do you think? Can you determine the probability that Mrs Müller has a son and a daughter?

Solution

In order to illustrate the solution, we choose an urn model. All families are considered, but we are interested in only those with two children. The urn model is limited to only those families with two children. These children could have been born in the following orders:

Girl, Girl (GG)

Girl, Boy (GB)

Boy, Boy (BB)

Boy, Girl (BG)

As the probability of giving birth to a girl or a boy is equal, the same number of each of these four family varieties is represented in the urn. Supposing

10,000 families with two daughters

10,000 families getting first a daughter, and then a son

10,000 families with two sons

10,000 families getting first a son, and then a daughter

Since we already know that Mrs Schmidt has a daughter, we take all families with two boys out of the urn. Now, it only contains families of the varieties GG, BG and GB – each variety is represented with the same number. In our example this means 10,000 families with two girls and a total number of 20,000 families having a son and a daughter. Therefore, the probability that Mrs Müller has two daughters is one-third.

Conversely, with a two-thirds probability, Mrs Müller has a son and a daughter.